VALUE PROBLEMS ASSOCIATED

COMPUTATION

OFGREEN’S MATRICESFOR BOUNDARY

VALUE PROBLEMS ASSOCIATED

WITH

A PAIR

OF

MIXED LINEAR REGULAR ORDINARY

DIFFERENTIAL OPERATORS

T.GNANA BHASKAR and M.VENKATESULU

Department

of Mathematics SriSathyaSaiInstituteofHigher Learning

Prasanthinilayam,515 134 INDIA

(Received

July29, 1991and inrevisedform

January

15,

1992)

ABSTRACT. An

algorithm^{for the}computationofGreen’smatricesforboundary^value

problems

associated withapairof mixedlinearregular ordinarydifferential operators is

presented

^andtwo examples fromthe studiesofacousticwaveguides inoceanand transverse vibrations innonho- mogeneous stringsare discussed.

KEY

WORDS AND PHRASES: Nonexplicitly mixed, matchingly mixed, boundary value

problem,

^{Green’smatrices.}

1991 AMS

SUBJECT CLASSIFICATION

CODES: 34XX, 34BXX, 34B27 1.

INTRODUCTION

Recently,anewclassof problemsof thetypewhere different differentialoperatorsare defined over twoadjacentintervals,involvingcertain mixed

(interface)

^{conditionsarestudied in}

[1,2,3,4].

Theseproblemsinvolveapairofdifferentialoperators of the type1ul _ko0

PeDu ^ul,

^defined

on the interval

J-[a,b]

^andx2u2-

Q,D’u2 u2

^{defined ontheadjacent interval}

^J2-[b,c],

-

^{<a<b<+% where:kisanunknown}k-0 ^constant(eigenvalue) ^andthe functions

u

^and

u-z

^are

requiredtosatisfycertain mixed conditionsatthe interfacex b.

In

mostofthecases, thecomplete setof physicalconditions onthe systemgivesrisetoselfadjoint eigenvalue problemsassociated withthepair 0q,%). Basedonthe interface conditions theseproblemscanbe classified intothree types,namely

(i)

where the valuesof

u

^and

u-z

^{are notexplicitlyrelatedtoeachotheratx b,}

(ii)

where

u

^and

u-z

^{arerequiredtosatisfy}the continuityconditions atx b, and

(iii)

where

u

^and

satisfycertainmatchingconditionsatx b.

The methodspresentedin

[4]

for the construction ofGreen’smatricesforthe

boundary

value

problems (BVPs)

^{associatedwith}

0:a, x2)

aretheoretical innatureand involvelengthycalculations.

Here,

in thispaperwepresent

(i)

simpler algorithmsfor thecomputation of Green’smatricesfor theBVPsassociated with

0:,),

^and

(ii)

constructtheGreen’smatricesfor the

problems

found in somephysicalsituations.

Before indicatingthe division of the work into sections,weintroduceafewnotations andmake someassumptions. For any compactintervalJof

R

andforanonnegative integer k,let

C(J)

denote thespaceof all k-timescontinuouslydifferentiablecomplexvalued functions defined on

J.

Fora functionu,letu⁾denote the

j

derivativeofu,if it exists. Foracompactinterval

J

of

R

andfora positivecontinuous(weight)function

r(x)

definedon

J,

let

L(J,r)

denote the Hilbertspaceof all

Lebesguemeasurablecomplexvalued functionsudefinedon

J

suchthat

r(x)[u(x)[

^isintegrable

overJ.

The innerproductinL(J,r) isgiven by

(u, v)-fu(x)v-r(x)dx,

^u,v

L(J,r),

where

v(x’-’-

denotes thecomplex conjugate of

v(x),

and the normisgiven by

I111- ^{Cx)luCx)l d} . ^eL2(J,r)

Let H(J,r)

denotethosefunctions

inACff)

such thatbothuandu⁾are inL

ff,r). t

C denotes thek-dimensionalcomplex spacewhoseelemenmwetaketobecolumnvectors. For

k

^{x matrix}

Adenote the inverse of awithcomplexentries,square

A"

^denotematrixtheA,xif it exists.

k

^{matrixwhich is}If

V

^andthe

V

^{conjugatearevector}transpose ofA.^spaces

^(over

^{the same}

t A -

field),

^then

V

^x

V

denotes the cartesianproductof

V

^and

V

taken in that order. Foranoperator

T, D(T),R(T),N(T), (T)

denote the domain,range,nullspaceand the dimensionofthe nullspace of

T,

respectively.

t X- L ff,r)

^x

L (J r)

bethe caesianproductHilbertspace equippedwiththe innerproduct

(-)

^{and the no}

"11

^{given by}

<{u,. }, {v,, v}> <,, v,>

+

<u.v, {,, u}. {,, v} ex,

and

{.,,.}II (II.,II

⁺

II.II’)’=, {.,,}

Let H H’(Jl, rl) H"(J2,r2)

be the cartesianproductBanachspace.

’- ^PD

^and

ASSUMPTION

1.

LetJl-[a,b]andJ2-[b,c],-<a

<b<c <+.

Letx,-,, _{._}

0 ^Q’D’

^betwoformaldifferential expressions, where

P . ^C(J),

^{k O,1 n,P,(x)}

^,,

⁰

on

J1; Q C(J2),

k O,1 m,Q,,,(x) 0 on

J2;

and

r,(x) . ^C(J,)

^and

^{r:,(x) C(J)}

^arepositive

realvalued functions. Wealso assumen m.

ASSUMPTION

2.

LetA

andB hem nandre tnmatrices with

complex

entries,respectively such that therange ofA rangeof

B,

andhence, rank ofA rankofB-m.

In

Section 1, we shall collecttogetherafewdefinitionsand results, fromour earlierpapers, whichwerequirehere.

In

Section 2,weshall presentalemmaregardingtheformofsolutionsof atypeof initial valueproblems

(IVPs)

associated with thepair(x,,x:,),^intermsof Green’smatrices.

In

Section 3, we shall presentanalgorithm forthecomputationofGreen’smatricesforthe

BVPs

associated withthe pair

(x,,x:0- In

Section 4,weshallconstructtheGreen’smatricesfor problems encounteredinthestudiesof acoustic wave guidesin oceanandtransversevibrations innonho- mogeneous strings.

2.

PRELIMINARIES

Letfbe

^acomplexvalued function definedonJ. Let

f -f/J,

^{1,2. LetJ}

-J LIJ2.

^Consider

(,,)u -/" (2.)

and the corresponding homogeneous equation

(,,’r,2)u

^0.

(2.2)

DEFINITION I. We

call a

complex

valuedfunction

u(x)

definedontheinterval

J,

a solution (nonexplicitly

mixed)

^of

(2.1)

if

(i)

the functions

u/J u ^AC"(J)

^and

u/J u ^AC"(J)

(ii) u

^and

u

^{satisfytheequations}xul

f,

^{for x}

J1

^{a.e., and}:2u2

f2,

for^x

J2

^a.e.,

^respec- DEFINITION

2.Wecallacomplexvalued function

u(x)

^definedontheinterval

J,

acontinuous solutionof

(2.1)

^if

(i)

u is asolutionof

(2.1)

in the senseof Definition 1, and

(ii)

the functions_uiandu satisfythecontinuityconditions atthe interfacepointx b,namely,

u)(b

-)=

u)(b

+), j O, m 1.

DEFINITION

3. Wecallacomplexvalued function

u(x)

defined onthe interval

J,

amatching solutionof

(2.1)

^if

(i)

uis asolution of

(2.1)

in the sense ofDefinition 1, and

(ii)

the functions

u

^and

u

^{satisfy certain matchingconditions atthe}interface pointx-b, namely, A

a _{(b) B} t2(b),

where

t2

t(b) column(ul(b), u)(b) u" ^-X)(b)),

and

t’-(b

z(b column(uz(b ),ut2X)(b

,u2

))

REMARK

1. All the above definitions can becarriedovertoequation

(2.2)

^also.

Below,we recallafew definitionsfrom

[6],

in the form, requiredhere.

Let

x

DEFINITION

5. Thenonexplicitlymixedoperator

N0:)

is defined

by

D(N(’r,))-- {{ut, u,} H/Bi’V({u,u2})-

^{O, n}

+m},

() {u, u ^(u,/,

where

-1 m-1

B({u,u2} )- o(a,u’(a)+ou)(b))+ o(?,iu’(b)+,iu’(c))

^{i-1 n +m}

arethe linearly independent nonexplicitlymixed

bounda

^values.

DEFITION

6. Thecontinuouslymixed operator

C()

isdefinedby

O(C(x))-{{u,u:} eH/Bf({u,})-O,

^i-1 n,

u)(b)-u2)(b), i-1,...,n}, c(x) {ut,} {xu,x})

where

,{u,u-

N-1

,u><a+O,u><)+6,u’<)

^i-

^L...,n

.0

arethe linearly independent continuouslymixed

bounda

^values.

DEFION

7.

e

matchinglymixedoperator

M(x)

^{is defined}by

where

D(M(x))- {{u,u2} eH/Bf({u,u})-O,

^{1 n+m,}

Aa(b)-B(b),}

M() {

ul,

u}

BiC{u,u:,})

N-1

,,oCa,u’)(a)

⁺

^bou’)(c))

⁺

,

^{a(b i-1 n,}

are thelinearly independent matchinglymixedboundaryvalues.

REMARK

2. Forthe sake of brevity,weshallstudy onlytheoperatorsN(r)andM(r)and the results for the operatorC(r)followbytakingA-B -I

(the

^{n n} identity

matrix)

^intheresults for

M(r).

ASSUMPTION

3. Forthematchinglymixed case weassumethatn m.

2.

LEMMA

REGARDING

THE IVPs ASSOCIATED WITH O:,r)

Weconsideraparticular type ofinitialvalueproblemassociated with

(rt, r2)

for nonexplicitly mixed andmatchinglymixedoperators andgivearesult aboutthe form of the solutionofthe

IVPs,

intermsofGreen’smatrices.

(I) NonexplicitlyMixed InitialValue Problems

Let _ul u,,l and _u12 u,a be functions inH(J,r) ^and

Hm(J,r2)

which form bases for the solution spaces of

rut-0

^and r2u2-0, respectively. Then, the pairs

{ul,0}, {u,0} {u,,,0}, {0,ul2} {0,u,,a} (all

^ofwhichbelongto

H)

^formbasisfor the solutionspace

ofN(r){u,u2}

-0

(for

^theexplicitform of thebasis see

[3]).

Define

N(z)

tobethe operator in

H

^suchthat

NCr) {{u,u2} H/u()(a)=O,

^{j -0} n 1,

u)Cb)-O,

j-0 m

1},

REMARK

3. Wenotethat theWronskianof_ul u,,, namely, W(Ull

Unl)(S

^{0for all}

s J1,and the Wronskianofu:,l,

u,,,:,

namely

W(u2 urn2) (s)

0foralls

J2.

^And,wedenote

by

W(u, un)(s)

the determinant obtainedby replacingthe

’’

^{column in the} corresponding Wronskianby

(0,

⁰

1)

C’,i 1 n. Similarly,we define

W/(u12, u,,,2)(s).

(II)

Matchinglymixedinitialvalueproblems

Letthesetof pairs

{u.,ul} {u,l,u,,2}

^beabasisforthe solutionspace

ofMo(r) {ut, u}

0, where

Mo(r) {{u,u} H/Aa(b)-Bu2(b)}

M0() {u,u_} {u,}.

Also,define

M(r)

^tobe the operator in

H

such that

M(r) { {u, u} H/u)(a O,

j

O,

...,n

1,aa(b Bud(b)}, M(r) {u, u2} {ru,r.zu}.

Thelemmabelow follows fromthe variationof parameters formula.

LEMMA. (I)

^Thesolution

{u,u-z} ofN(r){u,u2} {f,j}

^{isof the}form

u(x)- {u(x),u(x)}

; G(x,s)f(s)r(s)ds

^x

J G(x,s)f2(s)r(s)ds

^x

_J2

where

w.(u u.,)(s) u,(x)

Also,wedefine

o,

w, Cu, u.)(s)

u,(x)

Gzz(X’S)’i-lQ,,,(s)WCulz

,u,,,2)(s)

andcallG^’vastheGreen’smatrixfor

N"0:).

where

and

Also,wedefine

a<s <x<b,

b<s <x<c,

II)

Thesolution

{ul, uz} ofM(z){u,u2} {f,f2}

^isof theform

u(x)- {u(x),u2(x)}

/ Gff(x,s)f(s)rx(s)ds

x

a(x,s)(s)rl(s)

^/

a,’(x,s)(s)r,(s)

W,(u ,u.3(s)

c,(x,s)-,..p.(s)W(u u.O(s)

W,(u. u.)(s) u,(x)

::

a<s <x<b,

a<s<b, b<x<c,

and callG^uastheGreen’smatrixfor

M(r,).

b^<$<x <c

G_u 0

G2ff

^G

(2.3)

(2.4)

3.

COMPUTATIONAL ALGORITHM

FOR

THE GREEN’S MATRICES FOR OPERATORS ASSOCIATED WITH

In

thissection,proceedingalongthelinesof

[5],

we present analgorithmfor thecomputation ofGreen’smatricesfor operatorsassociated with

(I)

Nonexplicitlymixedoperator: Consider

{fx,j} X.

Let u(x) {ux(x),uz(x)} (N0:))

^-x

(f,J}.

^Then

(see [4]), u(x) {Ul(X), U2(/)}

f. G(x,s)f(s)r(s)ds+ f G(x,s)f(s)r,(s)ds,

f G(x’s)fl(s)rl(s)ds+ f, G(x’s)f(s)r2(s)ds’

^x

J.

(3.1)

wedenote

and we call G

"

^{the Green’s matrix} for the operator

N(x.).

^Let

v(x)-{v(x),v2(x)}

(N(x)) - {A(x),f(x)}. By

^Theorem4

[1],

^wehave

rl(N(r))

n+m. Since,

{ui

vi,u2-

v2}

belongs tothe solutionspaceof

x{u,u}

^{-O,thereexistsscalars}

c cn

^{such that}

^u(x)- {u(x),u2(x)},

qu,(x)

+

v(x), E cn ^.iu2(x)

⁺

v(x) ^(3.3)

-1 i-I

Applyingtheboundaryvalue on

(3.3),

^wehave

B({u,u})

⁰ 1,2,...,n +m. That is,

B:({ ciuil(X",

.t "-t

c"/iui(x’})"-B:({v"v2}’

^(3.4)

But,

where

and

B:i- ,.i(c:tu ^")(a)

⁺

^i,u)(b))

^{1 n}

t.

?itu)(a

⁺

6itu)(b

^{1, m 1, n+ m}

Relation

(3.4)

^{can now}be written as,

c.,B:,

⁺

, c. /.,B, -Bf({vx, ^{v2}), I,}

^{...,n +m.}

i-I

Itcan be verified that the coefficient matrix of the

(n

+

m) (n

+

m)

linearsystem

(3.5)

ⁱⁿ

(n

+

m)

unknowns,isnonsingular.

Now,

bythe choice of

{vl, v2},

^wehave

-1 -1

where

and

(3.6)

m-1

-,C) Y_, ,E ^{(u. u.:O()}

^tokuttc

Clearly, N H"(J,r)

and

NH’(J,r).

Rewriting

(3.5),

^we have, for

LetB-(B/,),i-

^{1 n}

andB--(B),j-O,...,m

^{l- 1,...,n+m.}

^LetB -[B,B].

Itcanbeshown that

B

is anonsingularmatrix. Thatis,det

B

0. Consider the

(n

+in)

(n

+

m)

^linearsystem,for l-1,...,n+m,

B,’,{z,,,z,}

⁺

B{z,,/,),z,./,} ^{--{,}, (3.7)}

i-l

- ^{B,,B},i

^{1,. n} and j-1,2,, m where

B

^and

Wehaveby Cramer’s rule,

{z,(s),ziz(s)} -

B

are determinants obtainedbyreplacingthe

’

^and

^j’

^columns

inB andB,

^bythe column vectors

(t,...,,.,>)

^{and (g}

,+,),

respectively.

at

is, each of

z(s)

and

z(s)

^{are linear} combinations

off,(s) andre(s),

respectively.

Hence, {z(s),z(s)}

H.

en,

wehaveby taking theinner-productof both sides of

(3.7)

with

{A,},

-1 -1

(by (3.6)),

^whichimplies that,

c ({zl(s),z(s)}, {ft(s),f2(s)}),

¹ n + m.

Combining

(3.3)

^and

(3.8),

^andcomparingwith

(5),

^{we get,}

G ^(x,s)-

Uil(X)Zil(S)

^a<x<s<b

62(x,s )- ui(xi(s),

^{a<x<b b}<s <c

(3.8)

G21(x,s)- ui(x)zi(s

), b<x <c a<s<b

i-I

[ ilui(x)z(s)+G(x,s),

^b<s<x<c

6(x,s)-

ui(x)zi(s

^b<x<s<c

This

completes

thealgorithm forthecomputation of

Green’s

matrixG^Nforthenonexplieitlymixed operator

N:).

REMARK

4. Thealgorithm for the computation of the

Green’s

matrixG^ufortheoperator

M(x),

^runsalongthe similar lines, withn m.

4. PHYSICAL

EXAMPLES

In

this section, we shall use thecomputational algorithms

developed

^{in Section}3,tocompute the Green’s matrices for a matchingly mixed operator an/] a continuously mixed operator, encountered in the studies of acousticwaveguidesin oceans andtransversevibrations innonho- mogeneous strings, respectively.

(I) Acoustic waveguidesinoceans [6]:

Considerthe oceantobeconsisting^oftwohomogeneous layers,with arigidbottom anda pressurerelease surface. Then, thepropagationof acousticwaveguidesin such an ocean^is

governed

bythefollowing equations.

r,u, u

^z)+

K2 ^{u Lug,}

^0<x<

d

+ <x<

togetherwiththe mixedboundaryconditionsgiven by,

u (0) o, u (clg

wherep_{and P2 are}constantdensities of thetwolayers, K,K2areconstantswhichdependuponthe frequencyconstant oand theconstantsound velocitiescl,c2ofthetwolayers, respectively,

.

^isan

unknown constant,

[0,d]

and

[dl, d2]

denotethedepth ofthetwolayersand

u

^andu2stand for the depth eigenfunctions.

LetJ [O,d]

^and

J2 [d,d2].

^Thematchingconditionsattheinterfacex

--dl

can be written in the matrix form

Axa(d)-B2a2(dt),

^where

ai(dO-column(ui(di),u.t,n(dO), Ai

_1/pi ^{for 1,2. Also,wehaven m d 2. Define}

M(x) { {u, u2} H2(J,

1/p)^x

H2(J2,1/p:)

/a

fl ^l(d)

^A

^fl(d,), u(O) u’)(d2) 0}, M(r,)u {’u,xzuz}

Aftersimplecalculationsalonglinesof thealgorithm,weget the formoftheGreen’smatrixG^uto beoftheform,

sinKx

KM (19.:,K cosK(d d) cosK(cl

^s

K ^sinK(cl

^s

sinK(cl d))

^{0 x<s}

d

G- _sinK:

[ ^KM

(PcsK(-d0csK(d-x)-pKsinK(d-x)sinK(-d0)’ 0<s<x

G -s,nKxx

^p’

cosK2(d2-s),

^0<x

<all, di

^<$<d

sinK: ^{cosK2(dz-x ),}

^0<s

^{<d, d}

^<x

^<d [osK(a-,)

G J|cosK2(d_x)(PzK

_sinK,

_d, csK2(d,-s)+P:,K, cosK,

^d

sinK(s -d,)), d,

^<s<x Wealsonotethat

and

AtdCdt, s) -A2G2t(dt,

^-M

^s)

AJ(d,,s AG(a,s

REMARK

5. Intheabove,wehave the compact andgeneralform of theGreen’smatrixof theproblem comparedtotheonegivenin

[6].

(II) Transversevibrationsinnonhomogeneous strings[7]:

Considerthestringconsisting of^twoportions^oflengths

dl

^and

d-

^{dl,anddifferentuniform}

densitiesp,_P2respectively,havingtensionTand stretchedbetweenthe pointsx 0andx d_2. The modesoftransversevibrationsofthe abovestringaregoverned by,

.- ^{c(-. ))} . ^o

^<x<

^d.

and

2. (2)

:2u2-c2t-u

)-Lu

2, d

l<x<d

^2,

togetherwiththe mixedboundaryconditionsgivenby,

u(0) u:Cd9 ^{0. u(4) u(d)} u)(d) u)(4)

where

c

^{T/pi, 1, 2.}

^Here,

the conditionsatthe interfacepointarethecontinuityconditions.

Proceeding alongthe lines of the algorithm,we get,afterroutinecalculations, the Green’s matrixGtobe of theform,

, 0<x<dl, d<s<d

, 0<x<s<d

, 0<s<x<d

Wenotethat

_,(dx,

s)-

t2C(dx,

s)and similarrelations are trueof thecomponents

G

^and

^G c.

ACKNOWLEDGMENT

Theauthorsdedicatethe worktothechancellor of theinstituteBhagavansrisathyasaibaba.

REFERENCES

[1] VENKATESULU,

M. and

BHASKAR,

T.

GNANA,

Solutions of initial value

problems

associatedwithapair^ofmixed linearordinarydifferentialequations, J.^Math.Anal_.Appl, 146

(2),

1991, 63-78.

[2] VENKATESULU,

M.and

BHASKAR,

T.QNANA,Fundamental systems and solutionsof nonhomogeneous equations associated with a pair of mixed linear ordinary differential equations, J.

Aust.

Math,

Soc,

seriesA49, 1990.

[3] VENKATESULU,

M.and

BHASKAR,

T.

GNANA,

Selfadjoint boundaryvalueproblems associated with apairof mixed linearordinarydifferentialequations,

J. Math.

Anal. Appl.

144

(2),

^1989.

[4] VENKATESULU,

M. and

BHASKAR,

T.

GNANA,

Green’smatricesfor boundary value

problems

associated with apair ofmixed linear

regular

ordinary differential operators, to appearin Zeitschriftf.Anal.

[5] LOCKER, JOHN,

Functionalanalysisandtwopointdifferential operators, Pitman Research

Notes,

1986.

[6] BOYLES,

C.

ALLAN, Acoustic

Waveguid.es, Applicationsto

O.ceani

^Sciences,Wiley, New York,1984.

[7] GHOSH,

P.

K...The

MathematicsofWavesand Vibrations, Macmillan, Delhi, India, 1975.